What is known is that there is an established tank with 3 fish in it. One fish was identified as a black moor and so we can easily see what size this fish can be.The two rainbows are a vague description- if they are a couple of pseudomugils they would produce a minimal bio-load but if they are something like boesemani (or bigger) the bio-load is much greater. So nobody responding to this thread, myself included, really knows for certain exactly how much bio-load the OP's tank can handle right now. However, I would bet the bows are not the tiny ones.
Since the OP said the plan is old fish out and new fish in over a relatively short time period, the only issue is do the old fish and the new fish generate similar bioloads. Tetras, even a dozen are light weight and the dwarf gouramis are not real big either. Not are huklis real big. The incoming bio-load is not huge for sure, even if all the fish are adult size.
My best estimate based on the minimal information presented is the trade off is closer to even than not. And if it isn't, I would bet the old fish produce more waste than the new fish. Goldfish are notoriously messy fish and none of the new ones are.
So, I would also bet that as long as the OP does nothing to trash the current bacteria in the system, they will be sufficient to handle all the replacement fish right away. Any shortfall should be minimal and at a low enough level and duration as not to be a serious concern.
To be on the safe side an ammonia test is a good idea as it would confirm what is up. I think this is the only kit needed as it will answer the question fast as to whether there is detectable ammonia or not once the new fish are in. It will also be useful down the road in case of other problems as it allows one to eliminate ammonia as a cause.
Joyboy- BBA hit it pretty much on the head. However, I would go even further and suggest that taking out the old fish, doing one decent water change and soon after putting in the new fish will be perfectly safe. I would say your original assumption that you have all the bacteria you need already is right on the money. If you are worried, then the water changes can't hurt, but then I would also suggest you feed very sparingly until the kits arrives. However, if you can get just the ammonia test kit quickly, you may be able to save yourself a lot of unneeded water changes and a few dollars on other kits as well